NONLINEAR DYNAMICS IN COUPLED FUZZY CONTROL-SYSTEMS .1. COHERENCE ANDCHAOS-FRUSTRATION IN TRIANGLE CONFIGURATION

Nonlinear dynamics and chaos are studied in a system for which a compl ete set of equations of motion such as equations of Newton, Navier-Sto kes and Van der Pol, is not available. As a very general system as suc h, we consider coupled classical spins (pendulums), each of which is u nder control by a fuzzy system that is designed to align the spin to a n unstable fixed point. The fuzzy system provides a deterministic proc edure to control an object without use of a differential equation. The positions and velocities of the spins are monitored periodically and each fuzzy control gives a momentum to its associated spin in the reve rse directions. If the monitoring is made with an interval short enoug h, the spin-spin interactions are overwhelmed by the fuzzy control and the system converges to a state as designed. However, a long-interval monitoring induces dynamics of ''too-late response'', and thereby res ults in chaos. A great variety of dynamics are generated under very de licate balance between the fuzzy control and the spin-spin interaction , in which two independent mechanisms of creating negative and positiv e ''Liapunov exponents'' interact with each other.